Express $z_1=-5\sqrt{5}-5\sqrt{15}i$ in polar form. Express your answer in exact terms, using radians, where your angle is between $0$ and $2\pi$ radians, inclusive. $z_1=$
The Strategy A complex number in rectangular form, $z={a}+{b}i$, can be written in polar form as $z={r}[\cos{\theta}+i\sin{\theta}]$, where ${r}$ is the absolute value, or modulus, and ${\theta}$ is the angle, or argument. Therefore, ${r}$ and ${\theta}$ can be found using the following formulas: ${r}=\sqrt{{a}^2+{b}^2}$ $\tan{\theta}=\dfrac{{b}}{{a}}$ [How did we get these equations?] Similarly, a complex number in polar form, $z={r}[\cos{\theta}+i\sin{\theta}]$, can be written in rectangular form as $z={a}+{b}i$, using the following formulas: ${a}={r}\cos{\theta}$ ${b}={r}\sin{\theta}$ [How did we get these equations?] Finding $r$ For $z_1={-5\sqrt{5}}{-5\sqrt{15}}i$ : ${a} = {-5\sqrt{5}}$ ${b} = {-5\sqrt{15}}$ Therefore, we can find ${r}$ as follows. $\begin{aligned}{r}&=\sqrt{{a}^2+{b}^2} \\\\&=\sqrt{({-5\sqrt{5}})^2+({-5\sqrt{15}})^2} \\\\&=\sqrt{125+375} \\\\&={\sqrt{500}} \\\\&={10\sqrt{5}}\end{aligned}$ Finding $\theta$ Using the formula, we have: $\begin{aligned}{\theta}&=\arctan\left(\dfrac{{b}}{{a}}\right) \\\\&=\arctan\left(\dfrac{{-5\sqrt{15}}}{{-5\sqrt{5}}}\right) \\\\&={\dfrac{\pi}{3}}\end{aligned}$ Since ${a}$ and ${b}$ are both negative, ${\theta}$ must lie in Quadrant $\text{III}$. Therefore its angle must be between $\pi$ and $\dfrac{3\pi}{2}$ radians. Using the identity $\tan(\pi+\theta)=\tan(\theta)$, we know that the following is also a solution of the equation. $\pi+\dfrac{\pi}{3}={\dfrac{4\pi}{3}}$ So $\theta = {\dfrac{4\pi}{3}}$. Summary $z_1={10\sqrt{5}}\left[\cos{\dfrac{4\pi}{3}}+i\sin{\dfrac{4\pi}{3}}\right]$